Surface Family with a Common Natural Asymptotic or Geodesic Lift of a Spacelike Curve with Spacelike Binormal in Minkowski 3-Space

Abstract

In the present paper, we find a surface family possessing the natural lift of a given spacelike curve with spacelike binormal as an asymptotic curve or geodesic in Minkowski 3-space. We express necessary and sufficient conditions for the given curve such that its natural lift is an asymptotic curve or geodesic on any member of the surface family. Finally, we illustrate the method with some examples.

Keywords

• Surface family,asymptotic curve,geodesic,natural lift

References

1. [1] G. J. Wang, K. Tang and C. L. Tai, Parametric representantion of a surface pencil with a common spatial geodesic, Comput. Aided Des., 36, (2004), 447–459.
2. [2] C. Y. Li, R. H. Wang and C. G. Zhu, Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Des., 43 (2011), 1110–1117.
3. [3] E. Bayram and E. Kasap, Hypersurface family with a common isoasymptotic curve, Geometry, No. 2014 (2014).
4. [4] E. Ergün, E. Bayram and E. Kasap, Surface pencil with a common line of curvature in Minkowski 3-space, Acta Mathematica Sinica-English Series, 30, 12 (2014), 2103-2118.
5. [5] Ergün, E., Bayram, E., and Kasap, E. Surface family with a common natural line of curvature lift. Journal of Science and Arts, no. 4(31) (2015).
6. [6] E. Ergün, E. Bayram and E. Kasap, Surface family with a common natural asymptotic lift, Journal of Science and Arts, no. 2(31) (2015), 117-124.
7. [7] E. Bayram and M. Bilici, Surface family with a common involute asymptotic curve, International Geometric Methods in Modern Physics, 13 (2016), 1650062.
8. [8] G. Şaffak Atalay and E. Kasap, Surfaces with common null asymptotic, Applied Mathematics and Computation, no. 260 (2015), 135-139.

9. [9] E. Kasap andF. T. Akyıldız, Surfaces with common geodesic in Minkowski 3-space, Appl. Math. Comput., 177 (2008), 260–270.
10. [10] G. Şaffak and E. Kasap, Family of surface with a common null geodesic, Int. J. Phys. Sci., Vol 4 8 (2009), 428–433.
11. [11] E. Bayram, E. Erg¨un and E. Kasap, Surface family with a common natural asymptotic lift, Journal of Science and Arts, No. 2(31) (2015), 117-124.
12. [12] E. Erg¨un and E. Bayram, Surface family with a common natural geodesic lift, Int. J. Math. Combin., Vol 1 (2016), 34-41.
13. [13] S. Kahyaoğlu and E. Kasap, Timelike minimal surface family prescribed by a spacelike curve in Minkowski 3-space, International Electronic Journal of Pure and Applied Mathematics, 10, 1 (2016).
14. [14] S. Kahyaoğlu and E. Kasap, Spacelike maximal surface family prescribed by a spacelike curve in 3-dimensional Minkowski space, International Journal of Contemporary Mathematical Sciences, 11, 3 (2016).
15. [15] S. Kahyaoğlu S and E. Kasap, An approach for minimal surface family passing a curve, International Journal of Contemporary Mathematical Sciences, 10, 5 (2015).
16. [16] O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
17. [17] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, 1994.
18. [18] M. Önder and H. H. Ug˘urlu, Frenet frames and invariants of timelike ruled surfaces, Ain Shams Engineering Journal, 4 (2013), 502-513.
19. [19] Walrave J. 1995, Curves and surfaces in Minkowski space, PhD. Thesis, K. U. Leuven Faculteit Der Wetenschappen.
20. [20] E. Erg¨un and M. C¸ alıs¸kan, On geodesic sprays In Minkowski 3-space, Int. J. Contem. Math. Sci., 6, 39 (2011), 1929-1933.